Beats, Fuzziness, and Tonality

Actually, Psycho-acoustics is one of the more fascinating area of audition, and I must admit that I use what I learned in those pre-dawn classes every day of my clinical life.

The question arises; assuming that the two notes are in the same or complimentary keys, when do two simultaneous notes  sound good together? Or conversely, when do two notes make the result sound fuzzy, and when do they seem to create a pulsing beating sound?

Music has all of these things, but speech never does. The lowest frequency component of speech is the fundamental frequency (f0) and this is typically above 100 Hz.  For women it may be nearer to 200 Hz, and for children, even higher.  In any event, it is never below 100 Hz.  There is minimal speech energy below the octave below middle C on a piano- speech is a right hand side of the piano keyboard phenomenon.  This means that the harmonics are never closer than 100 Hz.

In contrast, music uses the entire range of the piano keyboard, or at least “almost” the entire range. There are some composers who hit the very bottom key or two on the piano keyboard, but this is quite rare.  Harmonics, which in the case of a piano, are integer multiples of the fundamental frequency (f0) or as our musician colleagues say, “integer multiples of the tonic”.  For very low notes, the harmonics can be very close together, and in some cases, the harmonics of one note that is played at the same time as another note can indeed be within several Hz of another.  Also, unlike speech, in music, there are two (or more) notes that are played simultaneously and together make up a chord.

But let’s take a step back to a lecture on Psycho-acoustics: imagine a dark and dreary morning with a class full of sleepy eyed audiology students. The professor (who is also half asleep) tells us that if two notes are within about 20 Hz (or if any of their harmonics are within 20 Hz), then there is a creation of an additional very low frequency sound that seems to pulse at a well-defined frequency; this is referred to as “beats” and is a direct consequence of the difference in frequency (in Hz) between the two notes that are played at the same time- the lower the frequency of the beats, the closer are the two notes.

We actually see this all of the time since beats occur according to the DIFFERENCE between the two notes and not the absolute frequency of the two notes. If I am tuning my guitar and I want to make sure that a string is in tune (at least with the other strings on the guitar), I may have two notes- at 300 Hz and 310 Hz with one of them being just off tune with the other (by 10 Hz).  We would hear this as a slow beating and as we adjust the tuning of the rogue string, this beating slows down until it is no longer heard.  The string is now in tune.

Beats occur if the two simultaneous notes (or their harmonics) are within 20 Hz of each other and really other than tuning guitars, there is no musical function…. Actually that is not entirely true- my partner at the Musicians Clinics of Canada, Dr. John Chong, other than being an electrical engineer and a physician, has composed music for the National Ballet of Canada (when he was a mere teenager). And his composition used notes and harmonics that were very close to each other (less than 20 Hz) and the melody was defined by the frequency of the resultant beats.  And my partner is also a scratch golf player and I understand that you don’t want to be on the receiving end of his checks when he is playing hockey.

If the difference in two simultaneous notes (or their harmonics) is between 20 Hz and 30 Hz, then beats are not perceived but the difference sounds “fuzzy”. I can’t think of any musical reason to have notes that are only 20-30 Hz apart- it’s just not music anymore.  Well, that isn’t actually true either.

When I play the C major chord on my guitar, one cannot use the low E (the most bass string) yet, E is in the chord of C and we do see two other E notes being played for the C chord (on the second fret of the third string and on the top open string). Yet, if we add in the bass E note (that is, play all 6 strings during the C chord) it sounds quite bad.  Fancy guitarists use the bottom string but play the G note on the 3^rd fret of the bass E string.  G is also in the C chord but because it’s slightly higher in frequency than the bottom E, we can get away with it.

So, why can’t the bass E be played when it is indeed is one of the important elements in the C chord? Well, the bass E has a fundamental frequency that is very low and the guitar, being a half wavelength resonator instrument, has harmonics at integer multiples of the fundamental frequency.  And some of the harmonics of this low E are very close (but not the same) as the other elements in the C chord- too close to be considered to be “tonal”.  I am sure that even Schonberg- a major early 20^th century proponent of atonality- would cringe if he heard this bass E note being played in the C chord.  I haven’t seen Schonberg clinically recently, but I am sure that he would agree with me.

And if the difference between the frequencies of any two notes (or their harmonics) played simultaneously is greater than 30 Hz, then it is considered to have a tonal quality. Of course, the harmonics of one note may be so close (less than 30 Hz) from the harmonics of another note such that when the two notes are played together, sound awful- this is why we have different keys in music, and also why you don’t want to hear me sing.